Some statistics about tropical sandpile model

TY - JOUR

T1 - Some statistics about tropical sandpile model

AU - Kalinin, Nikita

AU - Prieto, Yulieth

N1 - Publisher Copyright: © 2023 Nikita Kalinin, Yulieth Prieto.

PY - 2023

Y1 - 2023

N2 - Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model repre-sents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of Z2. Given a set P of points in a compact convex domain Ω ⊂ R2 this linearized model produces a tropical polynomial GP 0Ω. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number n of randomly dropped points P = {p1, …, pn } ⊂ [0, 1]2 = Ω and the degree of the tropical polynomial GP 0Ω. We also study the distributions of the coefficients of GP 0Ω and the correlation between them. This paper’s main (experimental) result is that the tropical curve C(GP 0Ω) defined by GP 0Ω is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve C(GP 0Ω) are of directions (1, 0), (0, 1), (1, 1), (−1, 1). The main theoretical result is that C(GP 0Ω) \ (P ∩ ∂Ω), i.e. the tropical curve in Ω◦ with marked points P removed, is a tree.

AB - Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model repre-sents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of Z2. Given a set P of points in a compact convex domain Ω ⊂ R2 this linearized model produces a tropical polynomial GP 0Ω. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number n of randomly dropped points P = {p1, …, pn } ⊂ [0, 1]2 = Ω and the degree of the tropical polynomial GP 0Ω. We also study the distributions of the coefficients of GP 0Ω and the correlation between them. This paper’s main (experimental) result is that the tropical curve C(GP 0Ω) defined by GP 0Ω is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve C(GP 0Ω) are of directions (1, 0), (0, 1), (1, 1), (−1, 1). The main theoretical result is that C(GP 0Ω) \ (P ∩ ∂Ω), i.e. the tropical curve in Ω◦ with marked points P removed, is a tree.

KW - genus

KW - power law

KW - sandpile

KW - tropical geometry

UR - http://www.scopus.com/inward/record.url?scp=85167902553&partnerID=8YFLogxK

U2 - 10.46298/cm.10500

DO - 10.46298/cm.10500

M3 - 文章

AN - SCOPUS:85167902553

SN - 1804-1388

VL - 31

SP - 9

EP - 19

JO - Communications in Mathematics

JF - Communications in Mathematics

IS - 3

ER -